Hearing Protectors (ISO 4869-2)

A hearing protector is not measured on a coupler. ISO 4869-1 seats it on at least sixteen people and records the threshold shift each of them gets, so what comes back from the laboratory is a distribution: one attenuation per subject per octave band, with a spread that is often a third of the mean. ISO 4869-2 is the standard that turns that distribution into a number someone can act on, and the first thing it does is refuse to use the mean.

Every method starts from the assumed protection value, the mean attenuation reduced by a multiple of its own spread:

$\alpha$ is the inverse standard normal cumulative distribution at the protection performance $x$ (Table 1), so $AP{V}_{f84}$ with $\alpha =1$ is the attenuation 84 % of wearers reach or beat, and $AP{V}_{f98}$ with $\alpha =2$ is what all but one in fifty reach. Quoting a protector at its mean would describe a wearer who does not exist.

import numpy as np
from phonometry import hearing

# ISO 4869-1 attenuation of one protector: 16 subjects, eight octave bands
# from 63 Hz to 8 kHz. Annexes A to D of ISO 4869-2 work this same example
# through all three methods.
attenuation = np.array([
    [4, 8, 13, 18, 20, 30, 35, 30],   [6, 12, 16, 21, 29, 35, 47, 35],
    [10, 16, 17, 23, 25, 32, 48, 37], [3, 7, 12, 18, 20, 25, 33, 30],
    [8, 10, 16, 16, 25, 27, 43, 32],  [4, 7, 10, 15, 19, 32, 35, 31],
    [5, 5, 9, 16, 20, 25, 30, 28],    [15, 15, 21, 26, 25, 38, 46, 38],
    [5, 6, 10, 13, 19, 22, 29, 28],   [9, 9, 10, 19, 20, 27, 37, 31],
    [9, 16, 18, 24, 25, 35, 44, 39],  [5, 6, 11, 12, 17, 20, 28, 28],
    [7, 10, 17, 22, 25, 35, 41, 44],  [6, 8, 16, 18, 19, 19, 30, 33],
    [10, 12, 17, 25, 28, 33, 45, 40], [12, 13, 17, 27, 29, 38, 49, 41],
], dtype=float)

apv = hearing.assumed_protection_value(attenuation)          # x = 84 % by default
print(np.round(apv.mean_attenuation, 1))    # [ 7.4 10.  14.4 19.6 22.8 29.6 38.8 34.1]
print(np.round(apv.standard_deviation, 1))  # [3.3 3.6 3.6 4.6 4.  6.2 7.4 5.2]
print(np.round(apv.apv, 1))                 # [ 4.1  6.4 10.7 14.9 18.8 23.4 31.3 28.9]

# A stricter performance subtracts more of the same spread.
strict = hearing.assumed_protection_value(attenuation, performance=98)
print(np.round(apv.apv - strict.apv, 1))    # [3.3 3.6 3.6 4.6 4.  6.2 7.4 5.2]

Left, the protector: the assumed protection value sits a full standard deviation below the mean, and the gap is widest where the spread is, at 4 kHz. Right, the method: the HML line and the eight reference noises it was fitted on, with the three methods’ answers for the same noise.

import matplotlib.pyplot as plt

# apv is the AssumedProtectionResult computed above. One line:
apv.plot()   # mean, its spread shaded, and the assumed protection on top
plt.show()

# The HML side, by hand.
hml = hearing.hml_rating(attenuation)
high, medium, low = hml.reported
left, right = np.array([-4.0, 2.0]), np.array([2.0, 12.0])
fig, ax = plt.subplots()
ax.plot(left, medium - (high - medium) / 4 * (left - 2), color="#1f77b4")
ax.plot(right, medium - (medium - low) / 8 * (right - 2), color="#1f77b4")
ax.plot([-2, 2, 10], [high, medium, low], "o", color="#d62728", label="H, M, L")
differences = np.asarray(hearing.HML_REFERENCE_C_MINUS_A)
ax.plot(np.repeat(differences, 16), hml.predicted_reduction.T.reshape(-1), ".",
        color="#2ca02c", alpha=0.5, label="reference noises")
ax.set(xlabel="LpC - LpA [dB]", ylabel="Predicted noise level reduction [dB]")
ax.legend()
plt.show()

The most faithful, and the only one that sees the shape of the noise: subtract the assumed protection value band by band from the A-weighted spectrum and sum what is left.

# Annex B's noise: octave-band levels of a machine hall, LpA = 104 dB.
noise = [75.0, 84.0, 86.0, 88.0, 97.0, 99.0, 97.0, 96.0]
octave = hearing.octave_band_protected_level(noise, apv)
print(round(octave.effective_level, 1))   # 81.4
print(octave.reported_level)              # 81
print(round(octave.noise_reduction, 1))   # 22.6

The summation runs over the eight octaves from 63 Hz, or over seven from 125 Hz when either the noise or the protector has no 63 Hz data. Clause 6 rounds the result to the nearest integer, which is what reported_level does; effective_level keeps the unrounded value.

Three numbers instead of a spectrum. $H$, $M$ and $L$ are the predicted noise level reduction this protector gives for reference noises whose $(L_{p,C}-L_{p,A})$ is $-2$, $+2$ and $+10$ dB, fitted across the eight reference spectra of Table 2. Applying them needs only the C- and A-weighted levels of the real noise:

hml = hearing.hml_rating(attenuation)
print(hml.reported)                       # (24, 18, 13)

by_hml = hearing.hml_protected_level(104.0, 103.0, hml)
print(round(by_hml.noise_reduction, 1))   # 22.5
print(by_hml.reported_level)              # 82

Both branches meet at $+2$ dB, which is where $M$ itself is defined, so the line has a corner and no step. The values that enter them are the rounded ones: Clause 7.2 rounds $H$, $M$ and $L$ to the nearest integer, which is what a protector is published with, so that is what the application consumes.

One number, against a pink noise, subtracted from the C-weighted level.

snr = hearing.snr_rating(attenuation)
print(snr.reported)                       # 21

by_snr = hearing.snr_protected_level(snr, l_p_c=103.0)
print(by_snr.reported_level)              # 82

# When only the A-weighted level was recorded, Formula (24) reassembles the
# C-weighted one from an estimate of the difference and lands in the same place.
print(hearing.snr_protected_level(snr, l_p_a=104.0, c_minus_a=-1.0).reported_level)  # 82

Because the reference noise is fixed, the rating says nothing about the shape of the noise it will meet, which is exactly what the HML method’s three values recover.

The three methods answer the same question and rarely agree exactly. On the worked example above the same protector in the same noise gives 81 dB, 82 dB and 82 dB, and Clause 1’s own NOTE puts differences of 3 dB or less between comparable protectors below the resolution of the exercise. The ordering is not a ranking: the octave-band method uses more information and is the one to prefer when the spectrum is available, while HML and SNR exist precisely for when it is not.

The octave-band method starts at 63 Hz when both the noise and the protector have data there and at 125 Hz when either does not (Clause 6). The HML and SNR computations start at 125 Hz always, whatever is available at 63 Hz, which is why the reference spectra of Table 2 (Clause 7) and Table 3 (Clause 8) begin there.

One caution about the reference spectra: Annex C reprints Table 2 as its Table C.1 and the reprint disagrees with the original in two cells. Table 2 is the one that reproduces the annex’s own worked results, and it is the one this library carries; the discrepancy is registered in ERRATA.

• Covered

  ISO 4869-2:2018’s assumed protection value (Clause 5, Formula (1), with all seven protection performances of Table 1), the octave-band method (Clause 6, Formula (2)), the HML method (Clause 7, Formulae (3) to (18), including the eight reference noises and the empirical weights of Table 2) and the SNR method (Clause 8, Formulae (19) to (24), against the pink noise of Table 3), via hearing.assumed_protection_value, hearing.octave_band_protected_level, hearing.hml_rating, hearing.hml_protected_level, hearing.snr_rating and hearing.snr_protected_level. Every printed number of the worked example that runs through Annexes A to D is reproduced in the test suite and in the conformance report.

• Not covered

  The measurement that produces the attenuation values is ISO 4869-1’s and is not implemented: the real-ear attenuation at threshold, its subject panel, its fitting procedure and its own uncertainty are all taken as given, and this library starts from the resulting grid. ISO 4869-2’s Annex E uncertainty treatment of attenuation values and ratings is not implemented either. Nothing here models the difference between laboratory attenuation and what a protector achieves in the field, which is the subject of ISO/TR 4869-5 and is consistently large; a derating factor is a policy decision this library does not make for you.